Can a Change of Basis Matrix be Pre-multiplied with a Transformation Matrix?

[Sumanta]

Hello this is not a new thread but a continutaion of the thread of 6th april which I posted. I do not know how to continue the thread and so was not sure and so started as a new topic again. Please refer to my last post also.

This is regarding P(Tv) where T is the transfromation from V -> W and P is the change of basis from V -> V.


The points which siddharth etal mention is clear but what is unclear is the following What is unclear is that when u are doing this u are actually trying to premultiply a vector which is already in the space W. So does it mean if W ism dim space and P is n x n and m < n then when u multiply the P with Tv do u assume that the u extend the dimension of a vector in W to n by adding n -m 0 s to the end

 

[HallsofIvy]

What do you mean by "premultiply a vector which is already in the space W"? Whatever basis you use in V, you are multiplying a vector in V by a matrix to get a vector in W.
If V has dimension n and W dimension m, then, yes, P is n by n but T? is n by m: n columns and m rows. T(Pv)= (TP)v where v is in V and TP is also n by m.

 

[Sumanta]

Hi ,

What is understandable is the following

v in basis B -----------> v in basis B' -------------> W(on application of T)

v is in B and so Pv takes it to another basis in B'
so T(v) w.r.t basis B' is nothing but T (Pv) with respect to basis B.

q1. Can u apply T to a vector on another basis? Prob u can.

Is it P(Tv) or is it (PT)v that is being equated to T(Pv) ?

Probably second one. In that case I am not clear about the physical meaning of PT.


If not so then I am not clear what is the meaning of applying P on a vector in W since Tv is already a vector in W.

Thanks a lot for the help and putting up with me.

 

[matt grime]

This doesn't make any sense.

If P is a change of basis in V, you can't in general write down PT. since P is nxn and T is mxn. (V is n dim and W is m dim, and there is nothing that implies m=n).